convergent series

A series ∑an is said to be convergent
if the sequence of partial sums ∑i=1nai is convergent.
A series that is not convergent is said to be divergent.

A series ∑an is said to be absolutely convergent
if ∑|an| is convergent.

When the terms of the series live in ℝn,
an equivalent condition for absolute convergence of the series
is that all possible series obtained by rearrangements
of the terms are also convergent.
(This is not true in arbitrary metric spaces.)

It can be shown that absolute convergence implies convergence.
A series that converges, but is not absolutely convergent,
is called conditionally convergent.

Let ∑an be an absolutely convergent series,
and ∑bn be a conditionally convergent series.
Then any rearrangement of ∑an is convergent to the same sum.
It is a result due to Riemann that ∑bn
can be rearranged to converge to any sum, or not converge at all.